thank you for sharing this interesting question with the wefugees community.
let me try to answer your questions. It would be nice if someone could add:
1. You would then apply for a visa to reunite with your (German) child, therefore it is not necessary for your girlfriend to have any income.
2. You are Nigerian? The process is described by the German embassy in Lagos further:https://nigeria.diplo.de/blob/1426558/cd64eaa2de9468d5af2704d82d901d32/nachzug-zum-d-kind-data.pdf
The embassy states that the verification process of Nigerian Documents takes a t least 5 month...
3. The normal process is to go back and apply for family reunion from your country of origin. However, the Ausländerbehörde has the discretion (Ermessensspielraum) to decide to grant you the residence permit if you fulfill all the requirements and there are valid reasons for it (e.g. pregnant girlfriend, childcare...). The fact that you stayed illegal in Germany or that you entered Germany illegally might influence their discression. If Ausländerbehörde ties to sent you back to apply for a visa in your home country, the involvement of a lawyer for migration law might influence the discretion in your favor.
4. I don't think that the illegal stay will prevent the embassy from issuing the visa (if that is the only offense), maybe somebody else can answer this question more professional?
I hope my answers were helpful and I would be very happy if you would contact me again with further questions. I'm a consultant with mbeon and I‘ll be glad to help you over a longer period of time. Mbeon is an online advice for migrants via the mbeon app, which you can install on your smartphone for free. Download it from App Store or Google Play Store. You can find me in the mbeon app and get confidential individual advice in a secure virtual space without other people reading our conversation. Further information: ***.mbeon.de.